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I'm not sure if it's possible and I don't know the technical terms, but here goes.

These are two separate PNGs that I joined: one is a PNG of a brush stroke and the other is a simple container.

I would like to distort the brush stroke by adding a "curve" so that it fills the empty space of the container. enter image description here

The end-result would look something like this, but with the brushstroke PNG in the first screenshot: enter image description here

I currently have Adobe photoshop cs4, Fireworks cs4, and Illustrator cs4. Can I accomplish this with any of these apps?

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I guess you want soon also more complex forms than simple arc. This answer covers them.

Modern Illustrator could use your brush stroke PNG image as a brush - draw a curve with the Pen or other path making tool and apply your brush and that's that. Unfortunately legacy Illustrator doesn't accept such attempt with bitmap images, only vector shapes work as brush there.

I do not recommend as the first option tracing your PNG to vector because accurate enough look needs easily hundreds of small paths (=thousands of anchor points) which pull your computer on its knees if you want to paint something complex with that brush.

Test how good result your system can carry without slowing down too much. A few separate curves can well work ok, but painting with them can soon become very sluggish. See a tracing attempt in the end of this story.

There's available Envelope distortion which also can warp your PNG to the wanted geometric form. I recommend you it as the 1st option to test. An example:

enter image description here

This is your bush stroke clipped from your screenshot. In the next image it's imported to Illustrator and Object > Envelope Distortion > Make with Top Object is applied. The top object is the black closed path which can be quite random. But it must be a closed path:

enter image description here

Note that the empty area between the red stroke and the bounding box of the image take also their part of the room inside the envelope.

Read CS4 Illustrator's manual to read other envelope distortion options than Make with Top Object. A simple arc warping delivers what you asked in the question. Several simple warping options are also available in Photoshop.

Tracing and using as brush:

Here the brush stroke is traced to vector and the result (=about 3600 anchor points) is dragged to the Brushes collection and defined to be an art brush:

enter image description here

The brush is applied to a random path in the right. As a single path it doesn't degrade the apparent performance, but having them say 100 probably causes something noticeable. After having actually 128 copies In my old machine zooming and drawing something new started with about 0,5 seconds long pause.

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In Photoshop CS4 and later versions, as long as the paint stroke is on its own layer, you can use Edit > Transform > Warp, and move the curve handles/click and drag to distort as you wish.

Example

enter image description here

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